Difference between revisions of "Relationship between Chebyshev T and Gegenbauer C"

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==Theorem==
<strong>[[Relationship between Chebyshev T and Gegenbauer C|Theorem]]:</strong> The following formula holds:
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The following formula holds for $n \in \{1,2,3,\ldots\}$:
$$T_n(x)=\dfrac{n}{2} \displaystyle\lim_{\lambda \rightarrow 0} \dfrac{C_n^{\lambda}(x)}{\lambda}; n\geq 1,$$
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$$T_n(x)=\dfrac{n}{2} \displaystyle\lim_{\lambda \rightarrow 0} \dfrac{C_n^{\lambda}(x)}{\lambda},$$
where $T_n$ denotes a [[Chebyshev T]] polynomial and $C_n^{\lambda}$ denotes a [[Gegenbauer C]] polynomial.
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where $T_n$ denotes a [[Chebyshev T|Chebyshev polynomial of the first kind]] and $C_n^{\lambda}$ denotes a [[Gegenbauer C]] polynomial.
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<strong>Proof:</strong> █
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==Proof==
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==References==
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[[Category:Theorem]]
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[[Category:Unproven]]

Latest revision as of 22:33, 19 December 2017

Theorem

The following formula holds for $n \in \{1,2,3,\ldots\}$: $$T_n(x)=\dfrac{n}{2} \displaystyle\lim_{\lambda \rightarrow 0} \dfrac{C_n^{\lambda}(x)}{\lambda},$$ where $T_n$ denotes a Chebyshev polynomial of the first kind and $C_n^{\lambda}$ denotes a Gegenbauer C polynomial.

Proof

References